Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that is not cyclic.
Let (G = \langle g \rangle) be a cyclic group, and let (H) be a non-trivial subgroup of (G). Since (G) is cyclic, every element of (G) can be written as (g^n) for some integer (n). Because (H) is non-trivial, there exists a smallest __________ _______ __________ ________ _____ _______ _________ ______ __________.
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वह न्यूनतम दीजिए जिसके लिए An अनाबेली है। अपने उत्तर की पुष्टि कीजिए।
List two distinct cosets of in D10, where r is a reflection in D10.
. निम्नलिखित कथनों में से कौन-से कथन सत्य हैं? अपने उत्तरों की पुष्टि कीजिए।
i) यदि एक समूह G अपने किसी उचित उपसमूह के तुल्याकारी है, तो ।
ii) यदि x और y एक अना-बेली समूह (G, ) के ऐसे अवयव हैं जिनके लिए , तो
या
, जहाँ e, के सापेक्ष G का तत्समक है।
iii) अभाज्य कोटि वाला एक और केवल एक अनाबेली समूह है।
iv) यदि , जहाँ A एक समूह है, तो
।
v) यदि H और K समूह G के प्रसामान्य उपसमूह हैं, तो ।
Let be a fixed odd permutation in S10. Show that every odd permutation in S10 is a product of
and some permutation in A10.
यदि , तो o(g) अपरिमित है।
Which of the following statements are true? Give reasons for your answers.
i) If a group G is isomorphic to one of its proper subgroups, then .
ii) If x and y are elements of a non-abelian group (G, ) such that , then
or
, where e is the identity of G with respect to $$.
iii) There exists a unique non-abelian group of prime order.
iv) If , where A is a group, then
.
v) If H and K are normal subgroups of a group G, then .
Is , for any two ideals I and J of a ring R? Give reasons for your answer.